Group #21467445 — subgroups

Subgroups — C₃₁×#10686 #21467445

Order 7936.

410 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#1990	C₂	1	2	yes	maximal
2	C₃₁×#1956	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1844	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1834	C₂	1	2	yes	maximal
5	C₃₁×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×#106	—	2	2	no	—
8	C₂×C₃₁×#103	—	2	2	no	—
9	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#384	C₂²	1	1	yes	characteristic
12	C₃₁×#384	C₂²	1	2	yes	—
13	C₃₁×#340	—	2	4	no	—
14	C₃₁×#340	—	2	4	no	—
15	C₂²×Q₈×C₃₁	—	2	1	no	—
16	C₂²×C₈×C₃₁	D₄	1	4	yes	—
17	C₂²×C₈×C₃₁	D₄	1	2	yes	—
18	C₃₁×C₄⋊Q₈	—	2	2	no	—
19	C₃₁×C₄⋊Q₈	—	2	1	no	—
20	C₃₁×C₄⋊Q₈	Q₈	1	2	yes	—
21	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
22	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
23	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
24	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
28	C₃₁×#106	—	4	4	no	—
29	C₃₁×#103	—	4	4	no	—
30	C₄×C₈×C₃₁	—	2	2	no	—
31	C₄×C₈×C₃₁	—	2	2	no	—
32	C₄×C₈×C₃₁	—	2	2	no	—
33	C₂×Q₈×C₃₁	—	2	2	no	—
34	C₂×Q₈×C₃₁	—	4	1	no	—
35	C₂×Q₈×C₃₁	—	4	2	no	—
36	C₂×Q₈×C₃₁	—	4	2	no	—
37	C₂²×C₄×C₃₁	—	4	1	no	—
38	C₂²×C₄×C₃₁	—	4	1	no	—
39	C₂²×C₄×C₃₁	—	4	2	no	—
40	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
42	C₂×C₈×C₃₁	—	2	2	no	—
43	C₂×C₈×C₃₁	—	2	2	no	—
44	C₂×C₈×C₃₁	—	2	4	no	—
45	C₂×C₈×C₃₁	—	2	4	no	—
46	C₂×C₈×C₃₁	—	2	4	no	—
47	C₂×C₈×C₃₁	SD₁₆	1	4	yes	—
48	C₄⋊C₄×C₃₁	—	2	4	no	—
49	C₄⋊C₄×C₃₁	—	4	2	no	—
50	C₄⋊C₄×C₃₁	—	4	2	no	—
51	C₄⋊C₄×C₃₁	—	4	4	no	—
52	C₄⋊C₄×C₃₁	—	4	4	no	—
53	C₄⋊C₄×C₃₁	—	4	4	no	—
54	C₄⋊C₄×C₃₁	—	4	4	no	—
55	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₃₁	#41	1	1	yes	characteristic
57	C₄²×C₃₁	#41	1	2	yes	—
58	#10686	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₃₁	#127	1	1	yes	characteristic
60	Q₈×C₃₁	—	4	4	no	—
61	Q₈×C₃₁	—	8	2	no	—
62	C₂×C₄×C₃₁	—	4	2	no	—
63	C₂×C₄×C₃₁	—	8	1	no	—
64	C₂×C₄×C₃₁	—	8	1	no	—
65	C₂×C₄×C₃₁	—	8	1	no	—
66	C₂×C₄×C₃₁	—	8	2	no	—
67	C₂×C₄×C₃₁	—	8	2	no	—
68	C₂×C₄×C₃₁	—	8	2	no	—
69	C₂×C₄×C₃₁	—	8	2	no	—
70	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
72	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
73	C₂×C₄×C₃₁	#123	1	2	yes	—
74	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
75	C₂×C₄×C₃₁	#135	1	4	yes	—
76	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄×C₃₁	#137	1	4	yes	—
78	C₈×C₃₁	—	2	4	no	—
79	C₈×C₃₁	—	4	4	no	—
80	#1990	C₂×C₃₁	1	2	yes	—
81	#1956	C₂×C₃₁	1	1	yes	characteristic
82	#1844	C₂×C₃₁	1	1	yes	characteristic
83	#1834	C₂×C₃₁	1	2	yes	—
84	#1273	C₂×C₃₁	1	1	yes	characteristic
85	C₂²×C₃₁	#399	1	1	yes	characteristic
86	C₂²×C₃₁	#492	1	1	yes	characteristic
87	C₂²×C₃₁	#495	1	2	yes	—
88	C₂²×C₃₁	#496	1	1	yes	characteristic
89	C₂²×C₃₁	#497	1	2	yes	—
90	C₄×C₃₁	—	8	2	no	—
91	C₄×C₃₁	—	16	1	no	—
92	C₄×C₃₁	—	16	2	no	—
93	C₄×C₃₁	#476	1	2	yes	—
94	C₄×C₃₁	#479	1	4	yes	—
95	C₄×C₃₁	#481	1	2	yes	—
96	C₄×C₃₁	#485	1	4	yes	—
97	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
98	C₂×#106	—	2	2	no	—
99	C₂×#103	—	2	2	no	—
100	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
101	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
102	#384	C₂²×C₃₁	1	1	yes	characteristic
103	#384	C₂²×C₃₁	1	2	yes	—
104	#340	—	2	4	no	—
105	#340	—	2	4	no	—
106	C₂×C₃₁	#1708	1	1	yes	characteristic
107	C₂×C₃₁	#1716	1	1	yes	characteristic
108	C₂×C₃₁	#1719	1	2	yes	—
109	C₂×C₃₁	#2094	1	1	yes	characteristic
110	C₂×C₃₁	#2097	1	2	yes	—
111	C₂²×Q₈	—	2	1	no	—
112	C₂²×C₈	D₄×C₃₁	1	4	yes	—
113	C₂²×C₈	D₄×C₃₁	1	2	yes	—
114	C₄⋊Q₈	—	2	2	no	—
115	C₄⋊Q₈	—	2	1	no	—
116	C₄⋊Q₈	Q₈×C₃₁	1	2	yes	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	1	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
124	#106	—	4	4	no	—
125	#103	—	4	4	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₃₁	#10686	1	1	yes	characteristic, direct product factor, Sylow
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	4	1	no	—
132	C₂×Q₈	—	4	2	no	—
133	C₂×Q₈	—	4	2	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	SD₁₆×C₃₁	1	4	yes	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
153	C₄²	#41×C₃₁	1	1	yes	characteristic
154	C₄²	#41×C₃₁	1	2	yes	—
155	C₂³	C₃₁×#127	1	1	yes	characteristic
156	Q₈	—	4	4	no	—
157	Q₈	—	8	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
167	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
168	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
169	C₂×C₄	C₃₁×#123	1	2	yes	—
170	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
171	C₂×C₄	C₃₁×#135	1	4	yes	—
172	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
173	C₂×C₄	C₃₁×#137	1	4	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	4	4	no	—
176	C₂²	C₃₁×#399	1	1	yes	characteristic
177	C₂²	C₃₁×#492	1	1	yes	characteristic
178	C₂²	C₃₁×#495	1	2	yes	—
179	C₂²	C₃₁×#496	1	1	yes	characteristic
180	C₂²	C₃₁×#497	1	2	yes	—
181	C₄	—	8	2	no	—
182	C₄	—	16	1	no	—
183	C₄	—	16	2	no	—
184	C₄	C₃₁×#476	1	2	yes	—
185	C₄	C₃₁×#479	1	4	yes	—
186	C₄	C₃₁×#481	1	2	yes	—
187	C₄	C₃₁×#485	1	4	yes	—
188	C₂	C₃₁×#1708	1	1	yes	characteristic
189	C₂	C₃₁×#1716	1	1	yes	characteristic
190	C₂	C₃₁×#1719	1	2	yes	—
191	C₂	C₃₁×#2094	1	1	yes	characteristic
192	C₂	C₃₁×#2097	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#1990

C₂

yes

maximal

C₃₁×#1956

C₂

yes

maximal, characteristic

C₃₁×#1844

C₂

yes

maximal, characteristic

C₃₁×#1834

C₂

yes

maximal

C₃₁×#1273

C₂

yes

maximal, characteristic

C₂×C₃₁×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₃₁×#106

—

C₂×C₃₁×#103

—

C₂×C₄×C₈×C₃₁

C₂²

yes

—

C₂×C₄×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×#384

C₂²

yes

characteristic

C₃₁×#384

C₂²

yes

—

C₃₁×#340

—

C₃₁×#340

—

C₂²×Q₈×C₃₁

—

C₂²×C₈×C₃₁

D₄

yes

—

C₂²×C₈×C₃₁

D₄

yes

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

Q₈

yes

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₃₁×#106

—

C₃₁×#103

—

C₄×C₈×C₃₁

—

C₄×C₈×C₃₁

—

C₄×C₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

SD₁₆

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂×Q₈

yes

C₄²×C₃₁

yes

C₄²×C₃₁

yes

—

#10686

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

#127

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₃₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₃₁

#123

yes

characteristic

C₂×C₄×C₃₁

#123

yes

—

C₂×C₄×C₃₁

C₂×SD₁₆

yes

—

C₂×C₄×C₃₁

#135

yes

—

C₂×C₄×C₃₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₃₁

#137

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

#1990

C₂×C₃₁

yes

—

#1956

C₂×C₃₁

yes

characteristic

#1844

C₂×C₃₁

yes

characteristic

#1834

C₂×C₃₁

yes

—

#1273

C₂×C₃₁

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

—

C₂²×C₃₁

#496

yes

characteristic

C₂²×C₃₁

#497

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#476

yes

—

C₄×C₃₁

#479

yes

—

C₄×C₃₁

#481

yes

—

C₄×C₃₁

#485

yes

—

C₂×C₄⋊Q₈

C₂²×C₃₁

yes

characteristic

C₂×#106

—

C₂×#103

—

100

C₂×C₄×C₈

C₂²×C₃₁

yes

—

101

C₂×C₄×C₈

C₂²×C₃₁

yes

characteristic

102

#384

C₂²×C₃₁

yes

characteristic

103

#384

C₂²×C₃₁

yes

—

104

#340

—

105

#340

—

106

C₂×C₃₁

#1708

yes

characteristic

107

C₂×C₃₁

#1716

yes

characteristic

108

C₂×C₃₁

#1719

yes

—

109

C₂×C₃₁

#2094

yes

characteristic

110

C₂×C₃₁

#2097

yes

—

111

C₂²×Q₈

—

112

C₂²×C₈

D₄×C₃₁

yes

—

113

C₂²×C₈

D₄×C₃₁

yes

—

114

C₄⋊Q₈

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

Q₈×C₃₁

yes

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄²

C₂³×C₃₁

yes

characteristic

124

#106

—

125

#103

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₃₁

#10686

yes

characteristic, direct product factor, Sylow

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂×D₄×C₃₁

yes

characteristic

138

C₂²×C₄

C₂×D₄×C₃₁

yes

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

SD₁₆×C₃₁

yes

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄²

C₂×Q₈×C₃₁

yes

characteristic

153

C₄²

#41×C₃₁

yes

characteristic

154

C₄²

#41×C₃₁

yes

—

155

C₂³

C₃₁×#127

yes

characteristic

156

Q₈

—

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

C₃₁×C₂²⋊Q₈

yes

characteristic

167

C₂×C₄

C₃₁×C₂²⋊Q₈

yes

—

168

C₂×C₄

C₃₁×#123

yes

characteristic

169

C₂×C₄

C₃₁×#123

yes

—

170

C₂×C₄

C₂×SD₁₆×C₃₁

yes

—

171

C₂×C₄

C₃₁×#135

yes

—

172

C₂×C₄

C₃₁×AGL(1,ℤ/8)

yes

—

173

C₂×C₄

C₃₁×#137

yes

—

174

C₈

—

175

C₈

—

176

C₂²

C₃₁×#399

yes

characteristic

177

C₂²

C₃₁×#492

yes

characteristic

178

C₂²

C₃₁×#495

yes

—

179

C₂²

C₃₁×#496

yes

characteristic

180

C₂²

C₃₁×#497

yes

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

C₃₁×#476

yes

—

185

C₄

C₃₁×#479

yes

—

186

C₄

C₃₁×#481

yes

—

187

C₄

C₃₁×#485

yes

—

188

C₂

C₃₁×#1708

yes

characteristic

189

C₂

C₃₁×#1716

yes

characteristic

190

C₂

C₃₁×#1719

yes

—

191

C₂

C₃₁×#2094

yes

characteristic

192

C₂

C₃₁×#2097

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₃₁

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₃₁

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

N	H ≅ G/N	type
58. #10686	C₃₁	direct (N × H)
129. C₃₁	#10686	direct (N × H)

H ≅ G/N

type

58. #10686

C₃₁

direct (N × H)

129. C₃₁

#10686

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#1990	2
C₃₁×#1956	2
C₃₁×#1844	2
C₃₁×#1834	2
C₃₁×#1273	2
#10686	31

subgroup

index

C₃₁×#1990

C₃₁×#1956

C₃₁×#1844

C₃₁×#1834

C₃₁×#1273

#10686

Subgroups — C₃₁×#10686 #21467445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#10686 #21467445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10686 #21467445